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CONTENTS

List of Boxes xxvii

Preface xxxi

Acknowledgments xxxix

PART I FOUNDATIONS 1

1 Newtonian Physics: Geometric Viewpoint 5

1.1 Introduction 51.1.1 The Geometric Viewpoint on the Laws of Physics 51.1.2 Purposes of This Chapter 71.1.3 Overview of This Chapter 7

1.2 Foundational Concepts 81.3 Tensor Algebra without a Coordinate System 101.4 Particle Kinetics and Lorentz Force in Geometric Language 131.5 Component Representation of Tensor Algebra 16

1.5.1 Slot-Naming Index Notation 171.5.2 Particle Kinetics in Index Notation 19

1.6 Orthogonal Transformations of Bases 201.7 Differentiation of Scalars, Vectors, and Tensors; Cross Product and Curl 221.8 Volumes, Integration, and Integral Conservation Laws 26

1.8.1 Gauss’s and Stokes’ Theorems 271.9 The Stress Tensor and Momentum Conservation 29

1.9.1 Examples: Electromagnetic Field and Perfect Fluid 301.9.2 Conservation of Momentum 31

1.10 Geometrized Units and Relativistic Particles for Newtonian Readers 331.10.1 Geometrized Units 331.10.2 Energy and Momentum of a Moving Particle 34

Bibliographic Note 35

Track Two; see page xxxivNonrelativistic (Newtonian) kinetic theory; see page 96Relativistic theory; see page 96

vii

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2 Special Relativity: Geometric Viewpoint 37

2.1 Overview 372.2 Foundational Concepts 38

2.2.1 Inertial Frames, Inertial Coordinates, Events, Vectors, and Spacetime Diagrams 382.2.2 The Principle of Relativity and Constancy of Light Speed 422.2.3 The Interval and Its Invariance 45

2.3 Tensor Algebra without a Coordinate System 482.4 Particle Kinetics and Lorentz Force without a Reference Frame 49

2.4.1 Relativistic Particle Kinetics: World Lines, 4-Velocity, 4-Momentum andIts Conservation, 4-Force 49

2.4.2 Geometric Derivation of the Lorentz Force Law 522.5 Component Representation of Tensor Algebra 54

2.5.1 Lorentz Coordinates 542.5.2 Index Gymnastics 542.5.3 Slot-Naming Notation 56

2.6 Particle Kinetics in Index Notation and in a Lorentz Frame 572.7 Lorentz Transformations 632.8 Spacetime Diagrams for Boosts 652.9 Time Travel 67

2.9.1 Measurement of Time; Twins Paradox 672.9.2 Wormholes 682.9.3 Wormhole as Time Machine 69

2.10 Directional Derivatives, Gradients, and the Levi-Civita Tensor 702.11 Nature of Electric and Magnetic Fields; Maxwell’s Equations 712.12 Volumes, Integration, and Conservation Laws 75

2.12.1 Spacetime Volumes and Integration 752.12.2 Conservation of Charge in Spacetime 782.12.3 Conservation of Particles, Baryon Number, and Rest Mass 79

2.13 Stress-Energy Tensor and Conservation of 4-Momentum 822.13.1 Stress-Energy Tensor 822.13.2 4-Momentum Conservation 842.13.3 Stress-Energy Tensors for Perfect Fluids and Electromagnetic Fields 85


PART II STATISTICAL PHYSICS 91

3 Kinetic Theory 95

3.1 Overview 953.2 Phase Space and Distribution Function 97

3.2.1 Newtonian Number Density in Phase Space, N 973.2.2 Relativistic Number Density in Phase Space, N 99

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3.2.3 Distribution Function f (x , v , t) for Particles in a Plasma 1053.2.4 Distribution Function Iν/ν3 for Photons 1063.2.5 Mean Occupation Number η 108

3.3 Thermal-Equilibrium Distribution Functions 1113.4 Macroscopic Properties of Matter as Integrals over Momentum Space 117

3.4.1 Particle Density n, Flux S, and Stress Tensor T 1173.4.2 Relativistic Number-Flux 4-Vector �S and Stress-Energy Tensor TTT 118

3.5 Isotropic Distribution Functions and Equations of State 1203.5.1 Newtonian Density, Pressure, Energy Density, and Equation of State 1203.5.2 Equations of State for a Nonrelativistic Hydrogen Gas 1223.5.3 Relativistic Density, Pressure, Energy Density, and Equation of State 1253.5.4 Equation of State for a Relativistic Degenerate Hydrogen Gas 1263.5.5 Equation of State for Radiation 128

3.6 Evolution of the Distribution Function: Liouville’s Theorem, the CollisionlessBoltzmann Equation, and the Boltzmann Transport Equation 132

3.7 Transport Coefficients 1393.7.1 Diffusive Heat Conduction inside a Star 1423.7.2 Order-of-Magnitude Analysis 1433.7.3 Analysis Using the Boltzmann Transport Equation 144


4 Statistical Mechanics 155

4.1 Overview 1554.2 Systems, Ensembles, and Distribution Functions 157

4.2.1 Systems 1574.2.2 Ensembles 1604.2.3 Distribution Function 161

4.3 Liouville’s Theorem and the Evolution of the Distribution Function 1664.4 Statistical Equilibrium 168

4.4.1 Canonical Ensemble and Distribution 1694.4.2 General Equilibrium Ensemble and Distribution; Gibbs Ensemble;

Grand Canonical Ensemble 1724.4.3 Fermi-Dirac and Bose-Einstein Distributions 1744.4.4 Equipartition Theorem for Quadratic, Classical Degrees of Freedom 177

4.5 The Microcanonical Ensemble 1784.6 The Ergodic Hypothesis 1804.7 Entropy and Evolution toward Statistical Equilibrium 181

4.7.1 Entropy and the Second Law of Thermodynamics 1814.7.2 What Causes the Entropy to Increase? 183

4.8 Entropy per Particle 1914.9 Bose-Einstein Condensate 193

Contents ix



4.10 Statistical Mechanics in the Presence of Gravity 2014.10.1 Galaxies 2014.10.2 Black Holes 2044.10.3 The Universe 2094.10.4 Structure Formation in the Expanding Universe: Violent Relaxation

and Phase Mixing 2104.11 Entropy and Information 211

4.11.1 Information Gained When Measuring the State of a Systemin a Microcanonical Ensemble 211

4.11.2 Information in Communication Theory 2124.11.3 Examples of Information Content 2144.11.4 Some Properties of Information 2164.11.5 Capacity of Communication Channels; Erasing Information

from Computer Memories 216Bibliographic Note 218

5 Statistical Thermodynamics 219

5.1 Overview 2195.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics 221

5.2.1 Extensive and Intensive Variables; Fundamental Potential 2215.2.2 Energy as a Fundamental Potential 2225.2.3 Intensive Variables Identified Using Measuring Devices;

First Law of Thermodynamics 2235.2.4 Euler’s Equation and Form of the Fundamental Potential 2265.2.5 Everything Deducible from First Law; Maxwell Relations 2275.2.6 Representations of Thermodynamics 228

5.3 Grand Canonical Ensemble and the Grand-Potential Representationof Thermodynamics 229

5.3.1 The Grand-Potential Representation, and Computation of ThermodynamicProperties as a Grand Canonical Sum 229

5.3.2 Nonrelativistic van der Waals Gas 2325.4 Canonical Ensemble and the Physical-Free-Energy Representation

of Thermodynamics 2395.4.1 Experimental Meaning of Physical Free Energy 2415.4.2 Ideal Gas with Internal Degrees of Freedom 242

5.5 Gibbs Ensemble and Representation of Thermodynamics; Phase Transitionsand Chemical Reactions 246

5.5.1 Out-of-Equilibrium Ensembles and Their Fundamental Thermodynamic Potentialsand Minimum Principles 248

5.5.2 Phase Transitions 2515.5.3 Chemical Reactions 256

5.6 Fluctuations away from Statistical Equilibrium 260

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5.7 Van der Waals Gas: Volume Fluctuations and Gas-to-Liquid Phase Transition 2665.8 Magnetic Materials 270

5.8.1 Paramagnetism; The Curie Law 2715.8.2 Ferromagnetism: The Ising Model 2725.8.3 Renormalization Group Methods for the Ising Model 2735.8.4 Monte Carlo Methods for the Ising Model 279


6 Random Processes 283

6.1 Overview 2836.2 Fundamental Concepts 285

6.2.1 Random Variables and Random Processes 2856.2.2 Probability Distributions 2866.2.3 Ergodic Hypothesis 288

6.3 Markov Processes and Gaussian Processes 2896.3.1 Markov Processes; Random Walk 2896.3.2 Gaussian Processes and the Central Limit Theorem; Random Walk 2926.3.3 Doob’s Theorem for Gaussian-Markov Processes, and Brownian Motion 295

6.4 Correlation Functions and Spectral Densities 2976.4.1 Correlation Functions; Proof of Doob’s Theorem 2976.4.2 Spectral Densities 2996.4.3 Physical Meaning of Spectral Density, Light Spectra, and Noise

in a Gravitational Wave Detector 3016.4.4 The Wiener-Khintchine Theorem; Cosmological Density Fluctuations 303

6.5 2-Dimensional Random Processes 3066.5.1 Cross Correlation and Correlation Matrix 3066.5.2 Spectral Densities and the Wiener-Khintchine Theorem 307

6.6 Noise and Its Types of Spectra 3086.6.1 Shot Noise, Flicker Noise, and Random-Walk Noise; Cesium Atomic Clock 3086.6.2 Information Missing from Spectral Density 310

6.7 Filtering Random Processes 3116.7.1 Filters, Their Kernels, and the Filtered Spectral Density 3116.7.2 Brownian Motion and Random Walks 3136.7.3 Extracting a Weak Signal from Noise: Band-Pass Filter, Wiener’s Optimal Filter,

Signal-to-Noise Ratio, and Allan Variance of Clock Noise 3156.7.4 Shot Noise 321

6.8 Fluctuation-Dissipation Theorem 3236.8.1 Elementary Version of the Fluctuation-Dissipation Theorem; Langevin Equation,

Johnson Noise in a Resistor, and Relaxation Time for Brownian Motion 3236.8.2 Generalized Fluctuation-Dissipation Theorem; Thermal Noise in a

Laser Beam’s Measurement of Mirror Motions; Standard Quantum Limitfor Measurement Accuracy and How to Evade It 331

Contents xi



6.9 Fokker-Planck Equation 3356.9.1 Fokker-Planck for a 1-Dimensional Markov Process 3366.9.2 Optical Molasses: Doppler Cooling of Atoms 3406.9.3 Fokker-Planck for a Multidimensional Markov Process; Thermal Noise

in an Oscillator 343Bibliographic Note 345

PART III OPTICS 347

7 Geometric Optics 351

7.1 Overview 3517.2 Waves in a Homogeneous Medium 352

7.2.1 Monochromatic Plane Waves; Dispersion Relation 3527.2.2 Wave Packets 354

7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approximation andGeometric Optics 357

7.3.1 Geometric Optics for a Prototypical Wave Equation 3587.3.2 Connection of Geometric Optics to Quantum Theory 3627.3.3 Geometric Optics for a General Wave 3667.3.4 Examples of Geometric-Optics Wave Propagation 3687.3.5 Relation to Wave Packets; Limitations of the Eikonal Approximation

and Geometric Optics 3697.3.6 Fermat’s Principle 371

7.4 Paraxial Optics 3757.4.1 Axisymmetric, Paraxial Systems: Lenses, Mirrors, Telescopes, Microscopes,

and Optical Cavities 3777.4.2 Converging Magnetic Lens for Charged Particle Beam 381

7.5 Catastrophe Optics 3847.5.1 Image Formation 3847.5.2 Aberrations of Optical Instruments 395

7.6 Gravitational Lenses 3967.6.1 Gravitational Deflection of Light 3967.6.2 Optical Configuration 3977.6.3 Microlensing 3987.6.4 Lensing by Galaxies 401

7.7 Polarization 4057.7.1 Polarization Vector and Its Geometric-Optics Propagation Law 4057.7.2 Geometric Phase 406


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8 Diffraction 411

8.1 Overview 4118.2 Helmholtz-Kirchhoff Integral 413

8.2.1 Diffraction by an Aperture 4148.2.2 Spreading of the Wavefront: Fresnel and Fraunhofer Regions 417

8.3 Fraunhofer Diffraction 4208.3.1 Diffraction Grating 4228.3.2 Airy Pattern of a Circular Aperture: Hubble Space Telescope 4258.3.3 Babinet’s Principle 428

8.4 Fresnel Diffraction 4298.4.1 Rectangular Aperture, Fresnel Integrals, and the Cornu Spiral 4308.4.2 Unobscured Plane Wave 4328.4.3 Fresnel Diffraction by a Straight Edge: Lunar Occultation of a Radio Source 4328.4.4 Circular Apertures: Fresnel Zones and Zone Plates 434

8.5 Paraxial Fourier Optics 4368.5.1 Coherent Illumination 4378.5.2 Point-Spread Functions 4388.5.3 Abbe’s Description of Image Formation by a Thin Lens 4398.5.4 Image Processing by a Spatial Filter in the Focal Plane of a Lens: High-Pass,

Low-Pass, and Notch Filters; Phase-Contrast Microscopy 4418.5.5 Gaussian Beams: Optical Cavities and Interferometric Gravitational-Wave

Detectors 4458.6 Diffraction at a Caustic 451


9 Interference and Coherence 455

9.1 Overview 4559.2 Coherence 456

9.2.1 Young’s Slits 4569.2.2 Interference with an Extended Source: Van Cittert-Zernike Theorem 4599.2.3 More General Formulation of Spatial Coherence; Lateral Coherence Length 4629.2.4 Generalization to 2 Dimensions 4639.2.5 Michelson Stellar Interferometer; Astronomical Seeing 4649.2.6 Temporal Coherence 4729.2.7 Michelson Interferometer and Fourier-Transform Spectroscopy 4749.2.8 Degree of Coherence; Relation to Theory of Random Processes 477

9.3 Radio Telescopes 4799.3.1 Two-Element Radio Interferometer 4799.3.2 Multiple-Element Radio Interferometers 4809.3.3 Closure Phase 4819.3.4 Angular Resolution 482

Contents xiii



9.4 Etalons and Fabry-Perot Interferometers 4839.4.1 Multiple-Beam Interferometry; Etalons 4839.4.2 Fabry-Perot Interferometer and Modes of a Fabry-Perot Cavity

with Spherical Mirrors 4909.4.3 Fabry-Perot Applications: Spectrometer, Laser, Mode-Cleaning Cavity, Beam-

Shaping Cavity, PDH Laser Stabilization, Optical Frequency Comb 496

9.5 Laser Interferometer Gravitational-Wave Detectors 5029.6 Power Correlations and Photon Statistics: Hanbury Brown and Twiss Intensity

Interferometer 509Bibliographic Note 512

10 Nonlinear Optics 513

10.1 Overview 51310.2 Lasers 515

10.2.1 Basic Principles of the Laser 51510.2.2 Types of Lasers and Their Performances and Applications 51910.2.3 Ti:Sapphire Mode-Locked Laser 52010.2.4 Free Electron Laser 521

10.3 Holography 52110.3.1 Recording a Hologram 52210.3.2 Reconstructing the 3-Dimensional Image from a Hologram 52510.3.3 Other Types of Holography; Applications 527

10.4 Phase-Conjugate Optics 53110.5 Maxwell’s Equations in a Nonlinear Medium; Nonlinear Dielectric Susceptibilities;

Electro-Optic Effects 53610.6 Three-Wave Mixing in Nonlinear Crystals 540

10.6.1 Resonance Conditions for Three-Wave Mixing 54010.6.2 Three-Wave-Mixing Evolution Equations in a Medium That Is Dispersion-Free

and Isotropic at Linear Order 54410.6.3 Three-Wave Mixing in a Birefringent Crystal: Phase Matching and

Evolution Equations 54610.7 Applications of Three-Wave Mixing: Frequency Doubling, Optical Parametric

Amplification, and Squeezed Light 55310.7.1 Frequency Doubling 55310.7.2 Optical Parametric Amplification 55510.7.3 Degenerate Optical Parametric Amplification: Squeezed Light 556

10.8 Four-Wave Mixing in Isotropic Media 55810.8.1 Third-Order Susceptibilities and Field Strengths 55810.8.2 Phase Conjugation via Four-Wave Mixing in CS2 Fluid 55910.8.3 Optical Kerr Effect and Four-Wave Mixing in an Optical Fiber 562


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PART IV ELASTICITY 565

11 Elastostatics 567

11.1 Overview 56711.2 Displacement and Strain 570

11.2.1 Displacement Vector and Its Gradient 57011.2.2 Expansion, Rotation, Shear, and Strain 571

11.3 Stress, Elastic Moduli, and Elastostatic Equilibrium 57711.3.1 Stress Tensor 57711.3.2 Realm of Validity for Hooke’s Law 58011.3.3 Elastic Moduli and Elastostatic Stress Tensor 58011.3.4 Energy of Deformation 58211.3.5 Thermoelasticity 58411.3.6 Molecular Origin of Elastic Stress; Estimate of Moduli 58511.3.7 Elastostatic Equilibrium: Navier-Cauchy Equation 587

11.4 Young’s Modulus and Poisson’s Ratio for an Isotropic Material: A SimpleElastostatics Problem 589

11.5 Reducing the Elastostatic Equations to 1 Dimension for a Bent Beam: Cantilever Bridge,Foucault Pendulum, DNA Molecule, Elastica 592

11.6 Buckling and Bifurcation of Equilibria 60211.6.1 Elementary Theory of Buckling and Bifurcation 60211.6.2 Collapse of the World Trade Center Buildings 60511.6.3 Buckling with Lateral Force; Connection to Catastrophe Theory 60611.6.4 Other Bifurcations: Venus Fly Trap, Whirling Shaft, Triaxial Stars, and

Onset of Turbulence 60711.7 Reducing the Elastostatic Equations to 2 Dimensions for a Deformed Thin Plate:

Stress Polishing a Telescope Mirror 60911.8 Cylindrical and Spherical Coordinates: Connection Coefficients and Components

of the Gradient of the Displacement Vector 61411.9 Solving the 3-Dimensional Navier-Cauchy Equation in Cylindrical Coordinates 619

11.9.1 Simple Methods: Pipe Fracture and Torsion Pendulum 61911.9.2 Separation of Variables and Green’s Functions: Thermoelastic Noise

in Mirrors 622Bibliographic Note 627

12 Elastodynamics 629

12.1 Overview 62912.2 Basic Equations of Elastodynamics; Waves in a Homogeneous Medium 630

12.2.1 Equation of Motion for a Strained Elastic Medium 63012.2.2 Elastodynamic Waves 63612.2.3 Longitudinal Sound Waves 637

Contents xv



12.2.4 Transverse Shear Waves 63812.2.5 Energy of Elastodynamic Waves 640

12.3 Waves in Rods, Strings, and Beams 64212.3.1 Compression Waves in a Rod 64312.3.2 Torsion Waves in a Rod 64312.3.3 Waves on Strings 64412.3.4 Flexural Waves on a Beam 64512.3.5 Bifurcation of Equilibria and Buckling (Once More) 647

12.4 Body Waves and Surface Waves—Seismology and Ultrasound 64812.4.1 Body Waves 65012.4.2 Edge Waves 65412.4.3 Green’s Function for a Homogeneous Half-Space 65812.4.4 Free Oscillations of Solid Bodies 66112.4.5 Seismic Tomography 66312.4.6 Ultrasound; Shock Waves in Solids 663

12.5 The Relationship of Classical Waves to Quantum Mechanical Excitations 667Bibliographic Note 670

PART V FLUID DYNAMICS 671

13 Foundations of Fluid Dynamics 675

13.1 Overview 67513.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow Velocity;

Liquids versus Gases 67713.3 Hydrostatics 681

13.3.1 Archimedes’ Law 68413.3.2 Nonrotating Stars and Planets 68613.3.3 Rotating Fluids 689

13.4 Conservation Laws 69113.5 The Dynamics of an Ideal Fluid 695

13.5.1 Mass Conservation 69613.5.2 Momentum Conservation 69613.5.3 Euler Equation 69713.5.4 Bernoulli’s Theorem 69713.5.5 Conservation of Energy 704

13.6 Incompressible Flows 70913.7 Viscous Flows with Heat Conduction 710

13.7.1 Decomposition of the Velocity Gradient into Expansion, Vorticity, and Shear 71013.7.2 Navier-Stokes Equation 71113.7.3 Molecular Origin of Viscosity 71313.7.4 Energy Conservation and Entropy Production 714

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13.7.5 Reynolds Number 71613.7.6 Pipe Flow 716

13.8 Relativistic Dynamics of a Perfect Fluid 71913.8.1 Stress-Energy Tensor and Equations of Relativistic Fluid Mechanics 71913.8.2 Relativistic Bernoulli Equation and Ultrarelativistic Astrophysical Jets 72113.8.3 Nonrelativistic Limit of the Stress-Energy Tensor 723


14 Vorticity 729

14.1 Overview 72914.2 Vorticity, Circulation, and Their Evolution 731

14.2.1 Vorticity Evolution 73414.2.2 Barotropic, Inviscid, Compressible Flows: Vortex Lines Frozen into Fluid 73614.2.3 Tornados 73814.2.4 Circulation and Kelvin’s Theorem 73914.2.5 Diffusion of Vortex Lines 74114.2.6 Sources of Vorticity 744

14.3 Low-Reynolds-Number Flow—Stokes Flow and Sedimentation 74614.3.1 Motivation: Climate Change 74814.3.2 Stokes Flow 74914.3.3 Sedimentation Rate 754

14.4 High-Reynolds-Number Flow—Laminar Boundary Layers 75714.4.1 Blasius Velocity Profile Near a Flat Plate: Stream Function and

Similarity Solution 75814.4.2 Blasius Vorticity Profile 76314.4.3 Viscous Drag Force on a Flat Plate 76314.4.4 Boundary Layer Near a Curved Surface: Separation 764

14.5 Nearly Rigidly Rotating Flows—Earth’s Atmosphere and Oceans 76614.5.1 Equations of Fluid Dynamics in a Rotating Reference Frame 76714.5.2 Geostrophic Flows 77014.5.3 Taylor-Proudman Theorem 77114.5.4 Ekman Boundary Layers 772

14.6 Instabilities of Shear Flows—Billow Clouds and Turbulence in the Stratosphere 77814.6.1 Discontinuous Flow: Kelvin-Helmholtz Instability 77814.6.2 Discontinuous Flow with Gravity 78214.6.3 Smoothly Stratified Flows: Rayleigh and Richardson Criteria

for Instability 784Bibliographic Note 786

15 Turbulence 787

15.1 Overview 78715.2 The Transition to Turbulence—Flow Past a Cylinder 789

Contents xvii



15.3 Empirical Description of Turbulence 79815.3.1 The Role of Vorticity in Turbulence 799

15.4 Semiquantitative Analysis of Turbulence 80015.4.1 Weak-Turbulence Formalism 80015.4.2 Turbulent Viscosity 80415.4.3 Turbulent Wakes and Jets; Entrainment; the Coanda Effect 80515.4.4 Kolmogorov Spectrum for Fully Developed, Homogeneous,

Isotropic Turbulence 81015.5 Turbulent Boundary Layers 817

15.5.1 Profile of a Turbulent Boundary Layer 81815.5.2 Coanda Effect and Separation in a Turbulent Boundary Layer 82015.5.3 Instability of a Laminar Boundary Layer 82215.5.4 Flight of a Ball 823

15.6 The Route to Turbulence—Onset of Chaos 82515.6.1 Rotating Couette Flow 82515.6.2 Feigenbaum Sequence, Poincare Maps, and the Period-Doubling Route to

Turbulence in Convection 82815.6.3 Other Routes to Turbulent Convection 83115.6.4 Extreme Sensitivity to Initial Conditions 832


16 Waves 835

16.1 Overview 83516.2 Gravity Waves on and beneath the Surface of a Fluid 837

16.2.1 Deep-Water Waves and Their Excitation and Damping 84016.2.2 Shallow-Water Waves 84016.2.3 Capillary Waves and Surface Tension 84416.2.4 Helioseismology 848

16.3 Nonlinear Shallow-Water Waves and Solitons 85016.3.1 Korteweg–de Vries (KdV) Equation 85016.3.2 Physical Effects in the KdV Equation 85316.3.3 Single-Soliton Solution 85416.3.4 Two-Soliton Solution 85516.3.5 Solitons in Contemporary Physics 856

16.4 Rossby Waves in a Rotating Fluid 85816.5 Sound Waves 862

16.5.1 Wave Energy 86316.5.2 Sound Generation 86516.5.3 Radiation Reaction, Runaway Solutions, and Matched Asymptotic

Expansions 869Bibliographic Note 874

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17 Compressible and Supersonic Flow 875

17.1 Overview 87517.2 Equations of Compressible Flow 87717.3 Stationary, Irrotational, Quasi-1-Dimensional Flow 880

17.3.1 Basic Equations; Transition from Subsonic to Supersonic Flow 88017.3.2 Setting up a Stationary, Transonic Flow 88317.3.3 Rocket Engines 887

17.4 1-Dimensional, Time-Dependent Flow 89117.4.1 Riemann Invariants 89117.4.2 Shock Tube 895

17.5 Shock Fronts 89717.5.1 Junction Conditions across a Shock; Rankine-Hugoniot Relations 89817.5.2 Junction Conditions for Ideal Gas with Constant γ 90417.5.3 Internal Structure of a Shock 90617.5.4 Mach Cone 907

17.6 Self-Similar Solutions—Sedov-Taylor Blast Wave 90817.6.1 The Sedov-Taylor Solution 90917.6.2 Atomic Bomb 91217.6.3 Supernovae 914


18 Convection 917

18.1 Overview 91718.2 Diffusive Heat Conduction—Cooling a Nuclear Reactor; Thermal Boundary

Layers 91818.3 Boussinesq Approximation 92318.4 Rayleigh-Benard Convection 92518.5 Convection in Stars 93318.6 Double Diffusion—Salt Fingers 937


19 Magnetohydrodynamics 943

19.1 Overview 94319.2 Basic Equations of MHD 944

19.2.1 Maxwell’s Equations in the MHD Approximation 94619.2.2 Momentum and Energy Conservation 95019.2.3 Boundary Conditions 95319.2.4 Magnetic Field and Vorticity 957

19.3 Magnetostatic Equilibria 95819.3.1 Controlled Thermonuclear Fusion 95819.3.2 Z-Pinch 960

Contents xix



19.3.3 �-Pinch 96219.3.4 Tokamak 963

19.4 Hydromagnetic Flows 96519.5 Stability of Magnetostatic Equilibria 971

19.5.1 Linear Perturbation Theory 97119.5.2 Z-Pinch: Sausage and Kink Instabilities 97519.5.3 The�-Pinch and Its Toroidal Analog; Flute Instability; Motivation

for Tokamak 97819.5.4 Energy Principle and Virial Theorems 980

19.6 Dynamos and Reconnection of Magnetic Field Lines 98419.6.1 Cowling’s Theorem 98419.6.2 Kinematic Dynamos 98519.6.3 Magnetic Reconnection 986

19.7 Magnetosonic Waves and the Scattering of Cosmic Rays 98819.7.1 Cosmic Rays 98819.7.2 Magnetosonic Dispersion Relation 98919.7.3 Scattering of Cosmic Rays by Alfven Waves 992


PART VI PLASMA PHYSICS 995

20 The Particle Kinetics of Plasma 997

20.1 Overview 99720.2 Examples of Plasmas and Their Density-Temperature Regimes 998

20.2.1 Ionization Boundary 99820.2.2 Degeneracy Boundary 100020.2.3 Relativistic Boundary 100020.2.4 Pair-Production Boundary 100120.2.5 Examples of Natural and Human-Made Plasmas 1001

20.3 Collective Effects in Plasmas—Debye Shielding and Plasma Oscillations 100320.3.1 Debye Shielding 100320.3.2 Collective Behavior 100420.3.3 Plasma Oscillations and Plasma Frequency 1005

20.4 Coulomb Collisions 100620.4.1 Collision Frequency 100620.4.2 The Coulomb Logarithm 100820.4.3 Thermal Equilibration Rates in a Plasma 101020.4.4 Discussion 1012

20.5 Transport Coefficients 101520.5.1 Coulomb Collisions 101520.5.2 Anomalous Resistivity and Anomalous Equilibration 1016

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20.6 Magnetic Field 101920.6.1 Cyclotron Frequency and Larmor Radius 101920.6.2 Validity of the Fluid Approximation 102020.6.3 Conductivity Tensor 1022

20.7 Particle Motion and Adiabatic Invariants 102420.7.1 Homogeneous, Time-Independent Magnetic Field and No Electric Field 102520.7.2 Homogeneous, Time-Independent Electric and Magnetic Fields 102520.7.3 Inhomogeneous, Time-Independent Magnetic Field 102620.7.4 A Slowly Time-Varying Magnetic Field 102920.7.5 Failure of Adiabatic Invariants; Chaotic Orbits 1030


21 Waves in Cold Plasmas: Two-Fluid Formalism 1033

21.1 Overview 103321.2 Dielectric Tensor, Wave Equation, and General Dispersion Relation 103521.3 Two-Fluid Formalism 103721.4 Wave Modes in an Unmagnetized Plasma 1040

21.4.1 Dielectric Tensor and Dispersion Relation for a Cold, Unmagnetized Plasma 104021.4.2 Plasma Electromagnetic Modes 104221.4.3 Langmuir Waves and Ion-Acoustic Waves in Warm Plasmas 104421.4.4 Cutoffs and Resonances 1049

21.5 Wave Modes in a Cold, Magnetized Plasma 105021.5.1 Dielectric Tensor and Dispersion Relation 105021.5.2 Parallel Propagation 105221.5.3 Perpendicular Propagation 105721.5.4 Propagation of Radio Waves in the Ionosphere; Magnetoionic Theory 105821.5.5 CMA Diagram for Wave Modes in a Cold, Magnetized Plasma 1062

21.6 Two-Stream Instability 1065Bibliographic Note 1068

22 Kinetic Theory of Warm Plasmas 1069

22.1 Overview 106922.2 Basic Concepts of Kinetic Theory and Its Relationship to Two-Fluid Theory 1070

22.2.1 Distribution Function and Vlasov Equation 107022.2.2 Relation of Kinetic Theory to Two-Fluid Theory 107322.2.3 Jeans’ Theorem 1074

22.3 Electrostatic Waves in an Unmagnetized Plasma: Landau Damping 107722.3.1 Formal Dispersion Relation 107722.3.2 Two-Stream Instability 107922.3.3 The Landau Contour 108022.3.4 Dispersion Relation for Weakly Damped or Growing Waves 1085

Contents xxi



22.3.5 Langmuir Waves and Their Landau Damping 108622.3.6 Ion-Acoustic Waves and Conditions for Their Landau Damping to Be Weak 1088

22.4 Stability of Electrostatic Waves in Unmagnetized Plasmas 109022.4.1 Nyquist’s Method 109122.4.2 Penrose’s Instability Criterion 1091

22.5 Particle Trapping 109822.6 N-Particle Distribution Function 1102

22.6.1 BBGKY Hierarchy 110322.6.2 Two-Point Correlation Function 110422.6.3 Coulomb Correction to Plasma Pressure 1107


23 Nonlinear Dynamics of Plasmas 1111

23.1 Overview 111123.2 Quasilinear Theory in Classical Language 1113

23.2.1 Classical Derivation of the Theory 111323.2.2 Summary of Quasilinear Theory 112023.2.3 Conservation Laws 112123.2.4 Generalization to 3 Dimensions 1122

23.3 Quasilinear Theory in Quantum Mechanical Language 112323.3.1 Plasmon Occupation Number η 112323.3.2 Evolution of η for Plasmons via Interaction with Electrons 112423.3.3 Evolution of f for Electrons via Interaction with Plasmons 112923.3.4 Emission of Plasmons by Particles in the Presence of a Magnetic Field 113123.3.5 Relationship between Classical and Quantum Mechanical Formalisms 113123.3.6 Evolution of η via Three-Wave Mixing 1132

23.4 Quasilinear Evolution of Unstable Distribution Functions—A Bump in the Tail 113623.4.1 Instability of Streaming Cosmic Rays 1138

23.5 Parametric Instabilities; Laser Fusion 114023.6 Solitons and Collisionless Shock Waves 1142


PART VII GENERAL RELATIVITY 1151

24 From Special to General Relativity 1153

24.1 Overview 115324.2 Special Relativity Once Again 1153

24.2.1 Geometric, Frame-Independent Formulation 115424.2.2 Inertial Frames and Components of Vectors, Tensors, and Physical Laws 115624.2.3 Light Speed, the Interval, and Spacetime Diagrams 1159

24.3 Differential Geometry in General Bases and in Curved Manifolds 116024.3.1 Nonorthonormal Bases 1161

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24.3.2 Vectors as Directional Derivatives; Tangent Space; Commutators 116524.3.3 Differentiation of Vectors and Tensors; Connection Coefficients 116924.3.4 Integration 1174

24.4 The Stress-Energy Tensor Revisited 117624.5 The Proper Reference Frame of an Accelerated Observer 1180

24.5.1 Relation to Inertial Coordinates; Metric in Proper Reference Frame; Transport Lawfor Rotating Vectors 1183

24.5.2 Geodesic Equation for a Freely Falling Particle 118424.5.3 Uniformly Accelerated Observer 118624.5.4 Rindler Coordinates for Minkowski Spacetime 1187


25 Fundamental Concepts of General Relativity 1191

25.1 History and Overview 119125.2 Local Lorentz Frames, the Principle of Relativity, and Einstein’s Equivalence Principle 119525.3 The Spacetime Metric, and Gravity as a Curvature of Spacetime 119625.4 Free-Fall Motion and Geodesics of Spacetime 120025.5 Relative Acceleration, Tidal Gravity, and Spacetime Curvature 1206

25.5.1 Newtonian Description of Tidal Gravity 120725.5.2 Relativistic Description of Tidal Gravity 120825.5.3 Comparison of Newtonian and Relativistic Descriptions 1210

25.6 Properties of the Riemann Curvature Tensor 121325.7 Delicacies in the Equivalence Principle, and Some Nongravitational Laws of Physics in

Curved Spacetime 121725.7.1 Curvature Coupling in the Nongravitational Laws 1218

25.8 The Einstein Field Equation 122125.8.1 Geometrized Units 1224

25.9 Weak Gravitational Fields 122425.9.1 Newtonian Limit of General Relativity 122525.9.2 Linearized Theory 122725.9.3 Gravitational Field outside a Stationary, Linearized Source of Gravity 123125.9.4 Conservation Laws for Mass, Momentum, and Angular Momentum in

Linearized Theory 123725.9.5 Conservation Laws for a Strong-Gravity Source 1238


26 Relativistic Stars and Black Holes 1241

26.1 Overview 124126.2 Schwarzschild’s Spacetime Geometry 1242

26.2.1 The Schwarzschild Metric, Its Connection Coefficients, and Its CurvatureTensors 1242

Contents xxiii



26.2.2 The Nature of Schwarzschild’s Coordinate System, and Symmetries of theSchwarzschild Spacetime 1244

26.2.3 Schwarzschild Spacetime at Radii r �M : The Asymptotically Flat Region 124526.2.4 Schwarzschild Spacetime at r ∼M 1248

26.3 Static Stars 125026.3.1 Birkhoff’s Theorem 125026.3.2 Stellar Interior 125226.3.3 Local Conservation of Energy and Momentum 125526.3.4 The Einstein Field Equation 125726.3.5 Stellar Models and Their Properties 125926.3.6 Embedding Diagrams 1261

26.4 Gravitational Implosion of a Star to Form a Black Hole 126426.4.1 The Implosion Analyzed in Schwarzschild Coordinates 126426.4.2 Tidal Forces at the Gravitational Radius 126626.4.3 Stellar Implosion in Eddington-Finkelstein Coordinates 126726.4.4 Tidal Forces at r = 0—The Central Singularity 127126.4.5 Schwarzschild Black Hole 1272

26.5 Spinning Black Holes: The Kerr Spacetime 127726.5.1 The Kerr Metric for a Spinning Black Hole 127726.5.2 Dragging of Inertial Frames 127926.5.3 The Light-Cone Structure, and the Horizon 127926.5.4 Evolution of Black Holes—Rotational Energy and Its Extraction 1282

26.6 The Many-Fingered Nature of Time 1293Bibliographic Note 1297

27 Gravitational Waves and Experimental Tests of General Relativity 1299

27.1 Overview 129927.2 Experimental Tests of General Relativity 1300

27.2.1 Equivalence Principle, Gravitational Redshift, and Global PositioningSystem 1300

27.2.2 Perihelion Advance of Mercury 130227.2.3 Gravitational Deflection of Light, Fermat’s Principle, and Gravitational

Lenses 130527.2.4 Shapiro Time Delay 130827.2.5 Geodetic and Lense-Thirring Precession 130927.2.6 Gravitational Radiation Reaction 1310

27.3 Gravitational Waves Propagating through Flat Spacetime 131127.3.1 Weak, Plane Waves in Linearized Theory 131127.3.2 Measuring a Gravitational Wave by Its Tidal Forces 131527.3.3 Gravitons and Their Spin and Rest Mass 1319

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27.4 Gravitational Waves Propagating through Curved Spacetime 132027.4.1 Gravitational Wave Equation in Curved Spacetime 132127.4.2 Geometric-Optics Propagation of Gravitational Waves 132227.4.3 Energy and Momentum in Gravitational Waves 1324

27.5 The Generation of Gravitational Waves 132727.5.1 Multipole-Moment Expansion 132827.5.2 Quadrupole-Moment Formalism 133027.5.3 Quadrupolar Wave Strength, Energy, Angular Momentum, and Radiation

Reaction 133227.5.4 Gravitational Waves from a Binary Star System 133527.5.5 Gravitational Waves from Binaries Made of Black Holes, Neutron Stars,

or Both: Numerical Relativity 134127.6 The Detection of Gravitational Waves 1345

27.6.1 Frequency Bands and Detection Techniques 134527.6.2 Gravitational-Wave Interferometers: Overview and Elementary

Treatment 134727.6.3 Interferometer Analyzed in TT Gauge 134927.6.4 Interferometer Analyzed in the Proper Reference Frame of the

Beam Splitter 135227.6.5 Realistic Interferometers 135527.6.6 Pulsar Timing Arrays 1355


28 Cosmology 1361

28.1 Overview 136128.2 General Relativistic Cosmology 1364

28.2.1 Isotropy and Homogeneity 136428.2.2 Geometry 136628.2.3 Kinematics 137328.2.4 Dynamics 1376

28.3 The Universe Today 137928.3.1 Baryons 137928.3.2 Dark Matter 138028.3.3 Photons 138128.3.4 Neutrinos 138228.3.5 Cosmological Constant 138228.3.6 Standard Cosmology 1383

28.4 Seven Ages of the Universe 138328.4.1 Particle Age 138428.4.2 Nuclear Age 138728.4.3 Photon Age 1392

Contents xxv



28.4.4 Plasma Age 139328.4.5 Atomic Age 139728.4.6 Gravitational Age 139728.4.7 Cosmological Age 1400

28.5 Galaxy Formation 140128.5.1 Linear Perturbations 140128.5.2 Individual Constituents 140628.5.3 Solution of the Perturbation Equations 141028.5.4 Galaxies 1412

28.6 Cosmological Optics 141528.6.1 Cosmic Microwave Background 141528.6.2 Weak Gravitational Lensing 142228.6.3 Sunyaev-Zel’dovich Effect 1428

28.7 Three Mysteries 143128.7.1 Inflation and the Origin of the Universe 143128.7.2 Dark Matter and the Growth of Structure 144028.7.3 The Cosmological Constant and the Fate of the Universe 1444


References 1449

Name Index 1473

Subject Index 1477

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